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        <h2 id="Outline"><a href="#Outline" class="headerlink" title="Outline"></a>Outline</h2><p>Introduction<br>FP-Tree data structure<br>Step 1: FP-Tree Construction<br>Step 2: Frequent Itemset Generation<br>Discussion</p>
<a id="more"></a>
<h2 id="Theory"><a href="#Theory" class="headerlink" title="Theory"></a>Theory</h2><h3 id="Introduction"><a href="#Introduction" class="headerlink" title="Introduction"></a>Introduction</h3><p>Apriori: uses a generate-and-test approach – generates candidate itemsets and tests if they are frequent</p>
<ul>
<li>Generation of candidate itemsets is expensive (in both space<br>and time)</li>
<li>Support counting is expensive<ul>
<li>Subset checking (computationally expensive)</li>
<li>Multiple Database scans (I/O)</li>
</ul>
</li>
</ul>
<p>FP-Growth: allows frequent itemset discovery without candidate itemset generation. Two step approach:</p>
<pre><code>- Build a compact data structure called the FP-tree
    - Built using 2 passes over the data-set.
- Extracts frequent itemsets directly from the FP-tree
    - Traversal through FP-Tree
</code></pre><h3 id="Core-Data-Structure-FP-Tree"><a href="#Core-Data-Structure-FP-Tree" class="headerlink" title="Core Data Structure: FP-Tree"></a>Core Data Structure: FP-Tree</h3><img src="/2017/08/31/FP-growth/markdown-img-paste-20170831132645268.png" alt="Figure: FP-Tree" title="Figure: FP-Tree">
<ul>
<li>Nodes correspond to items and have a counter (i)</li>
<li>FP-Growth reads 1 transaction at a time and maps it to a path (i)</li>
<li>Fixed order is used, so paths can overlap when transactions share items (when they have the same prefix ). (iii)</li>
<li>In this case, counters are incremented (iii)</li>
<li>Pointers are maintained between nodes containing the same item, creating singly linked lists (dotted lines) (ii)</li>
<li>The more paths that overlap, the higher the compression. FP-tree may fit in memory.</li>
<li>Frequent itemsets extracted from the FP-Tree.</li>
</ul>
<h3 id="Step-1-FP-Tree-Construction-Example"><a href="#Step-1-FP-Tree-Construction-Example" class="headerlink" title="Step 1: FP-Tree Construction (Example)"></a>Step 1: FP-Tree Construction (Example)</h3><p>FP-Tree is constructed using 2 passes over the data-set:</p>
<ul>
<li>Pass 1:<ul>
<li>Scan data and find support for each item.</li>
<li>Discard infrequent items.</li>
<li>Sort frequent items in decreasing order based on their support.<ul>
<li>For our example: a; b; c ; d ; e</li>
<li>Use this order when building the FP-Tree, so common prefixes can be shared.</li>
</ul>
</li>
</ul>
</li>
<li>Pass 2: construct the FP-Tree (see Figure: FP-Tree Construction)<ul>
<li>Read transaction 1: {a, b}<ul>
<li>Create 2 nodes a and b and the path null -&gt; a -&gt; b. Set counts of a and b to 1.</li>
</ul>
</li>
<li>Read transaction 2: {b, c, d}<ul>
<li>Create 3 nodes for b, c and d and the path null -&gt; b -&gt; c -&gt; d . Set counts to 1.</li>
<li>Note that although transaction 1 and 2 share b, the paths are disjoint as they don’t share a common prefix. Add the link between the b’s.</li>
</ul>
</li>
<li>Read transaction 3: {a, c, d, e}<ul>
<li>It shares common prefix item a with transaction 1 so the path for transaction 1 and 3 will overlap and the frequency count for node a will be incremented by 1. Add links between the c ‘s and d ‘s.</li>
</ul>
</li>
<li>Continue until all transactions are mapped to a path in the FP-tree.</li>
</ul>
</li>
</ul>
<img src="/2017/08/31/FP-growth/markdown-img-paste-20170831133546453.png" alt="Figure: FP-Tree Construction" title="Figure: FP-Tree Construction">
<h4 id="FP-Tree-size"><a href="#FP-Tree-size" class="headerlink" title="FP-Tree size"></a>FP-Tree size</h4><ul>
<li>The FP-Tree usually has a smaller size than the uncompressed data – typically many transactions share items (and hence prefixes).<ul>
<li>Best case scenario: all transactions contain the same set of items. 1 path in the FP-tree.</li>
<li>Worst case scenario: every transaction has a unique set of items (no items in common).<ul>
<li>Size of the FP-tree is at least as large as the original data.</li>
<li>Storage requirements for the FP-tree are higher – need to store the pointers between the nodes and the counters.</li>
</ul>
</li>
<li>The size of the FP-tree depends on how the items are ordered.<ul>
<li>Ordering by decreasing support is typically used but it does not always lead to the smallest tree (it’s a heuristic).</li>
</ul>
</li>
</ul>
</li>
</ul>
<h3 id="Step-2-Frequent-Itemset-Generation"><a href="#Step-2-Frequent-Itemset-Generation" class="headerlink" title="Step 2: Frequent Itemset Generation"></a>Step 2: Frequent Itemset Generation</h3><ul>
<li>FP-Growth extracts frequent itemsets from the FP-tree.</li>
<li>Bottom-up algorithm – from the leaves towards the root<ul>
<li>Divide and conquer: first look for frequent itemsets ending in e , then de , etc. . . then d , then cd , etc. . .</li>
</ul>
</li>
<li>First, extract prefix path sub-trees ending in an item(set). (hint: use the header_table) (see: Figure: the prefix path sub-tree ending in an item(set))</li>
<li>Each prefix path sub-tree is processed recursively to extract the frequent itemsets. Solutions are then merged.<ul>
<li>E.g. the prefix path sub-tree for e will be used to extract frequent itemsets ending in e , then in de , ce , be and ae , then in cde , bde , cde , etc. (see: Figure: the prefix path sub-tree for e)</li>
<li>Divide and conquer approach</li>
</ul>
</li>
</ul>
<img src="/2017/08/31/FP-growth/markdown-img-paste-20170831134908639.png" alt="Figure: the prefix path sub-tree ending in an item(set)" title="Figure: the prefix path sub-tree ending in an item(set)">
<img src="/2017/08/31/FP-growth/markdown-img-paste-20170831135437901.png" alt="Figure: the prefix path sub-tree for e" title="Figure: the prefix path sub-tree for e">
<h4 id="Conditional-FP-Tree"><a href="#Conditional-FP-Tree" class="headerlink" title="Conditional FP-Tree"></a>Conditional FP-Tree</h4><ul>
<li>The FP-Tree that would be built if we only consider transactions containing a particular itemset (and then removing that itemset from all transactions).</li>
</ul>
<p>Example: FP-Tree conditional on e.<br>To obtain the conditional FP-tree for e from the prefix sub-tree ending in e :</p>
<ul>
<li>Update the support counts along the prefix paths (from e ) to reflect the number of transactions containing e .<ul>
<li>b and c should be set to 1 and a to 2. (compare Figure: Conditional FP-Tree and Figure: the prefix path sub-tree ending in an item(set))</li>
</ul>
</li>
<li>Remove the nodes containing e – information about node e is no longer needed because of the previous step (see: Figure: Remove the nodes containing e)<ul>
<li>Remove infrequent items (nodes) from the prefix paths</li>
<li>E.g. b has a support of 1 (note this really means be has a support of 1). i.e. there is only 1 transaction containing b and e so be is infrequent – can remove b.</li>
<li>Question: why were c and d not removed?</li>
</ul>
</li>
<li>Use the the conditional FP-tree for e to find frequent itemsets ending in de , ce and ae<ul>
<li>Note that be is not considered as b is not in the conditional FP-tree for e .</li>
<li>For each of them (e.g. de ), find the prefix paths from the conditional tree for e , extract frequent itemsets, generate conditional FP-tree, etc… (recursive)</li>
<li>Example: e -&gt; de -&gt; ade (fd ; e g,fa; d ; e g are found to be frequent) see(Figure: recursive)</li>
</ul>
</li>
<li>Use the the conditional FP-tree for e to find frequent itemsets ending in de , ce and ae<ul>
<li>Example: e -&gt; ce (fc ; e g is found to be frequent) (see Figure: find frequent itemsets)</li>
</ul>
</li>
</ul>
<img src="/2017/08/31/FP-growth/markdown-img-paste-20170831140123690.png" alt="Figure: conditional FP-Tree" title="Figure: conditional FP-Tree">
<img src="/2017/08/31/FP-growth/markdown-img-paste-20170831140916275.png" alt="Figure: update the support counts" title="Figure: update the support counts">
<img src="/2017/08/31/FP-growth/markdown-img-paste-20170831141110455.png" alt="Figure: Remove the nodes containing e" title="Figure: Remove the nodes containing e">
<img src="/2017/08/31/FP-growth/markdown-img-paste-20170831144525681.png" alt="Figure: recursive" title="Figure: recursive">
<img src="/2017/08/31/FP-growth/markdown-img-paste-20170831144703477.png" alt="Figure: find frequent itemsets" title="Figure: find frequent itemsets">
<h3 id="Result"><a href="#Result" class="headerlink" title="Result"></a>Result</h3><p>Frequent itemsets found (ordered by sufix and order in which they are found).</p>
<h3 id="Discussion"><a href="#Discussion" class="headerlink" title="Discussion"></a>Discussion</h3><h4 id="Advantages-of-FP-Growth"><a href="#Advantages-of-FP-Growth" class="headerlink" title="Advantages of FP-Growth"></a>Advantages of FP-Growth</h4><ul>
<li>only 2 passes over data-set</li>
<li>“compresses” data-set</li>
<li>no candidate generation</li>
<li>much faster than Apriori</li>
</ul>
<h4 id="Disadvantages-of-FP-Growth"><a href="#Disadvantages-of-FP-Growth" class="headerlink" title="Disadvantages of FP-Growth"></a>Disadvantages of FP-Growth</h4><ul>
<li>FP-Tree may not fit in memory!!</li>
<li>FP-Tree is expensive to build<ul>
<li>Trade-off: takes time to build, but once it is built, frequent itemsets are read off easily.</li>
<li>Time is wasted (especially if support threshold is high), as the only pruning that can be done is on single items.</li>
<li>support can only be calculated once the entire data-set is added to the FP-Tree.</li>
</ul>
</li>
</ul>
<h2 id="Coding"><a href="#Coding" class="headerlink" title="Coding"></a>Coding</h2><h3 id="Core-Data-Structure-FP-Tree-1"><a href="#Core-Data-Structure-FP-Tree-1" class="headerlink" title="Core Data Structure: FP-Tree"></a>Core Data Structure: FP-Tree</h3><figure class="highlight python"><table><tr><td class="gutter"><pre><div class="line">1</div><div class="line">2</div><div class="line">3</div><div class="line">4</div><div class="line">5</div><div class="line">6</div><div class="line">7</div><div class="line">8</div><div class="line">9</div><div class="line">10</div><div class="line">11</div><div class="line">12</div><div class="line">13</div><div class="line">14</div><div class="line">15</div></pre></td><td class="code"><pre><div class="line"><span class="class"><span class="keyword">class</span> <span class="title">TreeNode</span><span class="params">(object)</span>:</span></div><div class="line">    <span class="function"><span class="keyword">def</span> <span class="title">__init__</span><span class="params">(self, name, occur, parent)</span>:</span></div><div class="line">        self.name = name</div><div class="line">        self.count = occur</div><div class="line">        self.nodelink = <span class="keyword">None</span></div><div class="line">        self.parent = parent</div><div class="line">        self.children = &#123;&#125;</div><div class="line"></div><div class="line">    <span class="function"><span class="keyword">def</span> <span class="title">inc</span><span class="params">(self, n_occur)</span>:</span></div><div class="line">        self.count += n_occur</div><div class="line"></div><div class="line">    <span class="function"><span class="keyword">def</span> <span class="title">disp</span><span class="params">(self, ind=<span class="number">1</span>)</span>:</span></div><div class="line">        print(<span class="string">'  '</span> * ind, self.name, <span class="string">' '</span>, self.count)</div><div class="line">        <span class="keyword">for</span> child <span class="keyword">in</span> self.children.values():</div><div class="line">            child.disp(ind + <span class="number">1</span>)</div></pre></td></tr></table></figure>
<h3 id="FP-Tree-Construction"><a href="#FP-Tree-Construction" class="headerlink" title="FP-Tree Construction"></a>FP-Tree Construction</h3><figure class="highlight python"><table><tr><td class="gutter"><pre><div class="line">1</div><div class="line">2</div><div class="line">3</div><div class="line">4</div><div class="line">5</div><div class="line">6</div><div class="line">7</div><div class="line">8</div><div class="line">9</div><div class="line">10</div><div class="line">11</div><div class="line">12</div><div class="line">13</div><div class="line">14</div><div class="line">15</div><div class="line">16</div><div class="line">17</div><div class="line">18</div><div class="line">19</div><div class="line">20</div><div class="line">21</div><div class="line">22</div><div class="line">23</div><div class="line">24</div><div class="line">25</div><div class="line">26</div><div class="line">27</div><div class="line">28</div><div class="line">29</div><div class="line">30</div><div class="line">31</div><div class="line">32</div><div class="line">33</div><div class="line">34</div><div class="line">35</div><div class="line">36</div><div class="line">37</div><div class="line">38</div><div class="line">39</div><div class="line">40</div><div class="line">41</div><div class="line">42</div><div class="line">43</div><div class="line">44</div><div class="line">45</div><div class="line">46</div></pre></td><td class="code"><pre><div class="line"><span class="function"><span class="keyword">def</span> <span class="title">construct_fp_tree</span><span class="params">(X, min_support)</span>:</span></div><div class="line">    <span class="function"><span class="keyword">def</span> <span class="title">update_tree</span><span class="params">(items, intree, header_table, count)</span>:</span></div><div class="line">        <span class="keyword">if</span> items[<span class="number">0</span>] <span class="keyword">in</span> intree.children:</div><div class="line">            intree.children[items[<span class="number">0</span>]].inc(count)</div><div class="line">        <span class="keyword">else</span>:</div><div class="line">            curr_item_node = TreeNode(items[<span class="number">0</span>], count, intree)</div><div class="line">            intree.children[items[<span class="number">0</span>]] = curr_item_node</div><div class="line">            <span class="keyword">if</span> header_table[items[<span class="number">0</span>]][<span class="number">1</span>] <span class="keyword">is</span> <span class="keyword">None</span>:</div><div class="line">                header_table[items[<span class="number">0</span>]][<span class="number">1</span>] = curr_item_node</div><div class="line">            <span class="keyword">else</span>:</div><div class="line">                update_header(header_table[items[<span class="number">0</span>]][<span class="number">1</span>], curr_item_node)</div><div class="line"></div><div class="line">        <span class="keyword">if</span> len(items) &gt; <span class="number">1</span>:</div><div class="line">            update_tree(items[<span class="number">1</span>::], intree.children[items[<span class="number">0</span>]], header_table, count)</div><div class="line"></div><div class="line"></div><div class="line">    <span class="function"><span class="keyword">def</span> <span class="title">update_header</span><span class="params">(node2test, target_node)</span>:</span></div><div class="line">        <span class="keyword">while</span> node2test.nodelink <span class="keyword">is</span> <span class="keyword">not</span> <span class="keyword">None</span>:</div><div class="line">            node2test = node2test.nodelink</div><div class="line">        node2test.nodelink = target_node</div><div class="line"></div><div class="line"></div><div class="line">    <span class="comment"># Pass 1</span></div><div class="line">    header_table = &#123;&#125;</div><div class="line">    <span class="keyword">for</span> trans <span class="keyword">in</span> X:</div><div class="line">        <span class="keyword">for</span> item <span class="keyword">in</span> trans:</div><div class="line">            header_table[item] = header_table.get(item, <span class="number">0</span>) + X[trans]  <span class="comment"># X[trans]</span></div><div class="line">    <span class="keyword">for</span> item, support <span class="keyword">in</span> list(header_table.items()):</div><div class="line">        <span class="keyword">if</span> support &lt; min_support:</div><div class="line">            <span class="keyword">del</span> header_table[item]</div><div class="line">    freqset = set(header_table.keys())</div><div class="line">    <span class="keyword">if</span> len(freqset) == <span class="number">0</span>:</div><div class="line">        <span class="keyword">return</span> <span class="keyword">None</span>, <span class="keyword">None</span></div><div class="line">    <span class="keyword">for</span> item, support <span class="keyword">in</span> header_table.items():</div><div class="line">        header_table[item] = [support, <span class="keyword">None</span>]</div><div class="line">    <span class="comment"># Pass 2: construct the FP-Tree</span></div><div class="line">    ret_tree = TreeNode(<span class="string">'Null Set'</span>, <span class="number">1</span>, <span class="keyword">None</span>)</div><div class="line">    <span class="keyword">for</span> transet, count <span class="keyword">in</span> X.items():</div><div class="line">        local_dataset = &#123;&#125;</div><div class="line">        <span class="keyword">for</span> item <span class="keyword">in</span> transet:</div><div class="line">            <span class="keyword">if</span> item <span class="keyword">in</span> freqset:</div><div class="line">                local_dataset[item] = header_table[item][<span class="number">0</span>]</div><div class="line">        <span class="keyword">if</span> local_dataset:</div><div class="line">            items_sorted = sorted(local_dataset, key=local_dataset.get, reverse=<span class="keyword">True</span>)</div><div class="line">            update_tree(items_sorted, ret_tree, header_table, count)</div><div class="line">    <span class="keyword">return</span> ret_tree, header_table</div></pre></td></tr></table></figure>
<h3 id="Frequent-Itemset-Generation"><a href="#Frequent-Itemset-Generation" class="headerlink" title="Frequent Itemset Generation"></a>Frequent Itemset Generation</h3><figure class="highlight python"><table><tr><td class="gutter"><pre><div class="line">1</div><div class="line">2</div><div class="line">3</div><div class="line">4</div><div class="line">5</div><div class="line">6</div><div class="line">7</div><div class="line">8</div><div class="line">9</div><div class="line">10</div><div class="line">11</div><div class="line">12</div><div class="line">13</div><div class="line">14</div><div class="line">15</div><div class="line">16</div><div class="line">17</div><div class="line">18</div><div class="line">19</div><div class="line">20</div><div class="line">21</div><div class="line">22</div><div class="line">23</div><div class="line">24</div><div class="line">25</div><div class="line">26</div><div class="line">27</div><div class="line">28</div><div class="line">29</div></pre></td><td class="code"><pre><div class="line"><span class="function"><span class="keyword">def</span> <span class="title">generate_freq_itemset</span><span class="params">(intree, freqitem, min_support, pre_fix, freq_itemset_list)</span>:</span></div><div class="line"></div><div class="line">    <span class="function"><span class="keyword">def</span> <span class="title">ascend_tree</span><span class="params">(leaf_node, prefix_path)</span>:</span></div><div class="line">        <span class="keyword">if</span> leaf_node.parent <span class="keyword">is</span> <span class="keyword">not</span> <span class="keyword">None</span>:</div><div class="line">            prefix_path.append(leaf_node.name)</div><div class="line">            ascend_tree(leaf_node.parent, prefix_path)</div><div class="line"></div><div class="line"></div><div class="line">    <span class="function"><span class="keyword">def</span> <span class="title">find_prefiex_path</span><span class="params">(base_path, tree_node)</span>:</span></div><div class="line">        cond_paths = &#123;&#125;</div><div class="line">        <span class="keyword">while</span> tree_node <span class="keyword">is</span> <span class="keyword">not</span> <span class="keyword">None</span>:</div><div class="line">            prefix_path = []</div><div class="line">            ascend_tree(tree_node, prefix_path)</div><div class="line">            <span class="keyword">if</span> prefix_path:</div><div class="line">                cond_paths[frozenset(prefix_path[<span class="number">1</span>:])] = tree_node.count</div><div class="line">            tree_node = tree_node.nodelink</div><div class="line">        <span class="keyword">return</span> cond_paths</div><div class="line"></div><div class="line">    freqitem_sorted = sorted(freqitem, key=freqitem.get, reverse=<span class="keyword">True</span>)</div><div class="line"></div><div class="line">    <span class="keyword">for</span> item <span class="keyword">in</span> freqitem_sorted:</div><div class="line">        new_freq_itemset = pre_fix.copy()</div><div class="line">        new_freq_itemset.add(item)</div><div class="line">        freq_itemset_list.append(new_freq_itemset)</div><div class="line">        cond_patt_base = find_prefiex_path(item, freqitem[item][<span class="number">1</span>])</div><div class="line">        cond_tree, head = construct_fp_tree(cond_patt_base, min_support)</div><div class="line"></div><div class="line">        <span class="keyword">if</span> head <span class="keyword">is</span> <span class="keyword">not</span> <span class="keyword">None</span>:</div><div class="line">            generate_freq_itemset(cond_tree, head, min_support, new_freq_itemset, freq_itemset_list)</div></pre></td></tr></table></figure>
<h3 id="其他参看实现"><a href="#其他参看实现" class="headerlink" title="其他参看实现"></a>其他参看实现</h3><p><a href="http://adataanalyst.com/machine-learning/fp-growth-algorithm-python-3/" target="_blank" rel="external">http://adataanalyst.com/machine-learning/fp-growth-algorithm-python-3/</a><br><a href="https://github.com/evandempsey/fp-growth" target="_blank" rel="external">https://github.com/evandempsey/fp-growth</a></p>

      
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              <div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#Outline"><span class="nav-number">1.</span> <span class="nav-text">Outline</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Theory"><span class="nav-number">2.</span> <span class="nav-text">Theory</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#Introduction"><span class="nav-number">2.1.</span> <span class="nav-text">Introduction</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Core-Data-Structure-FP-Tree"><span class="nav-number">2.2.</span> <span class="nav-text">Core Data Structure: FP-Tree</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Step-1-FP-Tree-Construction-Example"><span class="nav-number">2.3.</span> <span class="nav-text">Step 1: FP-Tree Construction (Example)</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#FP-Tree-size"><span class="nav-number">2.3.1.</span> <span class="nav-text">FP-Tree size</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Step-2-Frequent-Itemset-Generation"><span class="nav-number">2.4.</span> <span class="nav-text">Step 2: Frequent Itemset Generation</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#Conditional-FP-Tree"><span class="nav-number">2.4.1.</span> <span class="nav-text">Conditional FP-Tree</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Result"><span class="nav-number">2.5.</span> <span class="nav-text">Result</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Discussion"><span class="nav-number">2.6.</span> <span class="nav-text">Discussion</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#Advantages-of-FP-Growth"><span class="nav-number">2.6.1.</span> <span class="nav-text">Advantages of FP-Growth</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#Disadvantages-of-FP-Growth"><span class="nav-number">2.6.2.</span> <span class="nav-text">Disadvantages of FP-Growth</span></a></li></ol></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Coding"><span class="nav-number">3.</span> <span class="nav-text">Coding</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#Core-Data-Structure-FP-Tree-1"><span class="nav-number">3.1.</span> <span class="nav-text">Core Data Structure: FP-Tree</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#FP-Tree-Construction"><span class="nav-number">3.2.</span> <span class="nav-text">FP-Tree Construction</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Frequent-Itemset-Generation"><span class="nav-number">3.3.</span> <span class="nav-text">Frequent Itemset Generation</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#其他参看实现"><span class="nav-number">3.4.</span> <span class="nav-text">其他参看实现</span></a></li></ol></li></ol></div>
            

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                var titleInLowerCase = title.toLowerCase();
                var content = data.content.trim().replace(/<[^>]+>/g,"");
                var contentInLowerCase = content.toLowerCase();
                var articleUrl = decodeURIComponent(data.url);
                var indexOfTitle = [];
                var indexOfContent = [];
                // only match articles with not empty titles
                if(title != '') {
                  keywords.forEach(function(keyword) {
                    function getIndexByWord(word, text, caseSensitive) {
                      var wordLen = word.length;
                      if (wordLen === 0) {
                        return [];
                      }
                      var startPosition = 0, position = [], index = [];
                      if (!caseSensitive) {
                        text = text.toLowerCase();
                        word = word.toLowerCase();
                      }
                      while ((position = text.indexOf(word, startPosition)) > -1) {
                        index.push({position: position, word: word});
                        startPosition = position + wordLen;
                      }
                      return index;
                    }

                    indexOfTitle = indexOfTitle.concat(getIndexByWord(keyword, titleInLowerCase, false));
                    indexOfContent = indexOfContent.concat(getIndexByWord(keyword, contentInLowerCase, false));
                  });
                  if (indexOfTitle.length > 0 || indexOfContent.length > 0) {
                    isMatch = true;
                    hitCount = indexOfTitle.length + indexOfContent.length;
                  }
                }

                // show search results

                if (isMatch) {
                  // sort index by position of keyword

                  [indexOfTitle, indexOfContent].forEach(function (index) {
                    index.sort(function (itemLeft, itemRight) {
                      if (itemRight.position !== itemLeft.position) {
                        return itemRight.position - itemLeft.position;
                      } else {
                        return itemLeft.word.length - itemRight.word.length;
                      }
                    });
                  });

                  // merge hits into slices

                  function mergeIntoSlice(text, start, end, index) {
                    var item = index[index.length - 1];
                    var position = item.position;
                    var word = item.word;
                    var hits = [];
                    var searchTextCountInSlice = 0;
                    while (position + word.length <= end && index.length != 0) {
                      if (word === searchText) {
                        searchTextCountInSlice++;
                      }
                      hits.push({position: position, length: word.length});
                      var wordEnd = position + word.length;

                      // move to next position of hit

                      index.pop();
                      while (index.length != 0) {
                        item = index[index.length - 1];
                        position = item.position;
                        word = item.word;
                        if (wordEnd > position) {
                          index.pop();
                        } else {
                          break;
                        }
                      }
                    }
                    searchTextCount += searchTextCountInSlice;
                    return {
                      hits: hits,
                      start: start,
                      end: end,
                      searchTextCount: searchTextCountInSlice
                    };
                  }

                  var slicesOfTitle = [];
                  if (indexOfTitle.length != 0) {
                    slicesOfTitle.push(mergeIntoSlice(title, 0, title.length, indexOfTitle));
                  }

                  var slicesOfContent = [];
                  while (indexOfContent.length != 0) {
                    var item = indexOfContent[indexOfContent.length - 1];
                    var position = item.position;
                    var word = item.word;
                    // cut out 100 characters
                    var start = position - 20;
                    var end = position + 80;
                    if(start < 0){
                      start = 0;
                    }
                    if (end < position + word.length) {
                      end = position + word.length;
                    }
                    if(end > content.length){
                      end = content.length;
                    }
                    slicesOfContent.push(mergeIntoSlice(content, start, end, indexOfContent));
                  }

                  // sort slices in content by search text's count and hits' count

                  slicesOfContent.sort(function (sliceLeft, sliceRight) {
                    if (sliceLeft.searchTextCount !== sliceRight.searchTextCount) {
                      return sliceRight.searchTextCount - sliceLeft.searchTextCount;
                    } else if (sliceLeft.hits.length !== sliceRight.hits.length) {
                      return sliceRight.hits.length - sliceLeft.hits.length;
                    } else {
                      return sliceLeft.start - sliceRight.start;
                    }
                  });

                  // select top N slices in content

                  var upperBound = parseInt('1');
                  if (upperBound >= 0) {
                    slicesOfContent = slicesOfContent.slice(0, upperBound);
                  }

                  // highlight title and content

                  function highlightKeyword(text, slice) {
                    var result = '';
                    var prevEnd = slice.start;
                    slice.hits.forEach(function (hit) {
                      result += text.substring(prevEnd, hit.position);
                      var end = hit.position + hit.length;
                      result += '<b class="search-keyword">' + text.substring(hit.position, end) + '</b>';
                      prevEnd = end;
                    });
                    result += text.substring(prevEnd, slice.end);
                    return result;
                  }

                  var resultItem = '';

                  if (slicesOfTitle.length != 0) {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + highlightKeyword(title, slicesOfTitle[0]) + "</a>";
                  } else {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + title + "</a>";
                  }

                  slicesOfContent.forEach(function (slice) {
                    resultItem += "<a href='" + articleUrl + "'>" +
                      "<p class=\"search-result\">" + highlightKeyword(content, slice) +
                      "...</p>" + "</a>";
                  });

                  resultItem += "</li>";
                  resultItems.push({
                    item: resultItem,
                    searchTextCount: searchTextCount,
                    hitCount: hitCount,
                    id: resultItems.length
                  });
                }
              })
            };
            if (keywords.length === 1 && keywords[0] === "") {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-search fa-5x" /></div>'
            } else if (resultItems.length === 0) {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-frown-o fa-5x" /></div>'
            } else {
              resultItems.sort(function (resultLeft, resultRight) {
                if (resultLeft.searchTextCount !== resultRight.searchTextCount) {
                  return resultRight.searchTextCount - resultLeft.searchTextCount;
                } else if (resultLeft.hitCount !== resultRight.hitCount) {
                  return resultRight.hitCount - resultLeft.hitCount;
                } else {
                  return resultRight.id - resultLeft.id;
                }
              });
              var searchResultList = '<ul class=\"search-result-list\">';
              resultItems.forEach(function (result) {
                searchResultList += result.item;
              })
              searchResultList += "</ul>";
              resultContent.innerHTML = searchResultList;
            }
          }

          if ('auto' === 'manual') {
            input.addEventListener('input', inputEventFunction);
          } else {
            $('.search-icon').click(inputEventFunction);
            input.addEventListener('keypress', function (event) {
              if (event.keyCode === 13) {
                inputEventFunction();
              }
            });
          }

          // remove loading animation
          $(".local-search-pop-overlay").remove();
          $('body').css('overflow', '');

          proceedsearch();
        }
      });
    }

    // handle and trigger popup window;
    $('.popup-trigger').click(function(e) {
      e.stopPropagation();
      if (isfetched === false) {
        searchFunc(path, 'local-search-input', 'local-search-result');
      } else {
        proceedsearch();
      };
    });

    $('.popup-btn-close').click(onPopupClose);
    $('.popup').click(function(e){
      e.stopPropagation();
    });
    $(document).on('keyup', function (event) {
      var shouldDismissSearchPopup = event.which === 27 &&
        $('.search-popup').is(':visible');
      if (shouldDismissSearchPopup) {
        onPopupClose();
      }
    });
  </script>





  

  

  

  
  
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